Southern Cross is a lovely property, we've visited there and enjoyed a drink but we have never stayed there.
Most travelers to Little Cayman fly first to Grand Cayman and go through customs and immigration on GC before rechecking their bags and going through GCM Security to access the departure lounge and board their flight to Little Cayman. Each adult can bring a liter of duty-free liquor into Cayman, but remember to pack it carefully inside your checked luggage before rechecking it for your final flight. You won't be able to carry it through Security because it is larger than 3 oz. and doesn't fit inside a quart bag.
If you have a good interval before your flight from GCM to LCM (or on your return flight home from GCM) you can catch the free shuttle to the George Town Yacht Club, just outside the ticketing hall at the GC airport, to have a drink or a meal. The GTYC is located on the marina and the food is good, especially the cracked conch. They open at 11 am on weekdays but they are open early (8:30 am) on Saturdays and Sundays for breakfast. It's only about 3 minutes away and it is a very pleasant place to eat or drink, nicer than the airport, see the link below:
https://www.gtyachtclub.com/
You are allowed a total of 55 lbs luggage on Cayman Airways and that means everything, including carry-ons, and you may be charged per pound for overweight baggage. You won't be allowed any large carry-ons (no rollerboards or large back packs, etc.) because you will be flying a small Twin Otter plane. Large carry-ons will be placed in the baggage hold.
The LC airport is very tiny and it doesn't have any lights so the planes only fly during daylight hours. The runway is used as a road when there is no air traffic. Sometimes you fly a turboprop from GC to Brac and then transfer to a twin otter for the flight to LC - other times you fly an Otter directly from GC to LC. All the flights are short. On the day before you are scheduled to fly back to GC, you are usually allowed to pack a bag and send it back early. It will be waiting for you at the GC airport when you fly there the next day.
It is not unusual to have a bag delayed by a day. The last time we were there in 2017 the suitcase with most of our clothes was delayed for a day. Another couple had the bag with all their dive gear delayed - but the dive shop loaned them gear for free until their stuff arrived. I tend to bring a large purse that I can jam under the tiny seat in front of me and I pack it with essentials like medications, glasses and contact lenses, C-cards, swimsuits, and clean undies, phones and dive computers, just things that we will really need if our luggage is delayed. The staff at the LCBR laundered our clothes for free to help us get by until our missing bag arrived.
Little Cayman is a very small and quiet island. There's only one small village on the island and it's called Blossom Village. It has a grocery store, a liquor store, a car rental agency, and a gas pump and I think that's all the shops, but the grocery store (and liquor store?) is open on Sundays - at least it was the last time we were there.
The resort can arrange a rental car for you or you can arrange one yourself. You will have to pay $20 Cayman for a temporary drivers license unless you have an international license. The car rental agency can issue the temporary license if you show them a valid drivers license from back home. The license is valid on all 3 Cayman islands so if you are visiting multiple islands make sure they date it for your full trip.
There's very little traffic on LC so driving on the left is easy. Some folks like to rent motorbikes but it's a longish trip if you plan to drive around the island, stop at the village, the beach, the museum, the other resorts, etc., we prefer to rent a small car.
If you do rent a car for a day or two you may want to stop at the Hungry Iguana, it's a local restaurant and bar near the village and the lionfish tacos are very popular. It's also near Paradise Villas which is a small resort. It's not a dive-dedicated resort but I am sure that you can arrange diving if you stay about anywhere on the island. The last time we were on LC a local condo owner joined out boat with Reef Divers a few times.
The small LC Museum is also near the village and is a charming and interesting place to visit.
Damian, a former bartender at the Beach-Nuts Bar at the LCBR, now operates a glass-bottom boat and offers tours, fishing, and other water excursions. You might arrange an excursion with him over to uninhabited Owens Island to snorkel or fish. I believe that he also hosts a Lazy Sunday Barbecue on Owens Island that sounds like fun, see the link below:
https://www.caymancompass.com/2019/01/08/restored-glass-bottom-boat-now-little-cayman-tourist-vessel/
None of the LC resorts are large but the Little Cayman Beach Resort is the largest. They've just completed some renovations at LCBR and they are scheduled to open a underwater video and photography shop soon.There's a lovely little spa at the LCBR if you want to have a relaxing massage or other spa service, but book early because it fills up quickly, especially near the end of the week. There may be other spa offerings on the island, I don't know.
I'm not a fan of Karaoke night at the Beach-Nuts bar but Trivia night can be a lot of fun IMO. There's usually a barbecue on Friday nights and also a Sunday Brunch at LCBR. They also have a basketball and tennis court there.
In addition to SCC and LCBR, there's a 3rd dive resort on Little Cayman called Pirates Point and it is said to have the best food. They offer a prix-fixe gourmet menu but the choices are limited. We have never visited or eaten there, see link below:
http://www.piratespointresort.com/
The Salt Rock Nature Trail on LC is a popular nature-walk and hiking trail. It started as a footpath between Blossom Village and the Salt Pier.
There are many beautiful bird species on LC and the Booby Pond Nature Reserve supports the largest red-footed booby population in the Caribbean. You will see lots of Frigate Birds above the island also. LC also has a large, healthy population of the native iguana species.
The Central Caribbean Marine Institute (CCMI) is located on LC and in the past they have given tours of the facility on Friday afternoons but I don't know if they still do that, see link below:
https://reefresearch.org/
I think that Little Cayman has some of the best diving in the Caribbean. The reefs are beautiful and healthy and you will see lots of beautiful corals and sea life including eagle rays and friendly groupers. The reefs start shallow and they have a lot of swim-throughs and tunnels that you can either explore or swim over.
I have never done any shore diving on LC but
@Ricardo V. has and he shared this report about shore diving on Little Cayman:
https://www.scubaboard.com/community/posts/8139619/
There is a recompression chamber on the nearby island of Cayman Brac.
Reef Divers, the dive op at LCBR, will take a boat over to Cayman Brac to dive the Tibbetts wreck (the Russian Frigate) and that is a fun dive. I imagine that other dive ops on the island will do the same if enough divers are interested in the excursion.
Here's some links about LC:
http://www.alertdiver.com/Caymans_Sister_Islands
http://www.alertdiver.com/Sister_Islands_gallery
http://www.qsl.net/zf2nt/index.htm
https://www.caymancompass.com/2015/07/24/From-Grand-to-Little--a-relaxing-island-hop/
https://www.caymancompass.com/2015/08/07/little-cayman-s-big-summer-shutdown/
https://www.caribjournal.com/2017/06/22/why-you-need-to-visit-little-cayman/#
https://youtu.be/Q6pJ7-Acks0
https://youtu.be/QDb1nkROOcE
https://www.facebook.com/littlecaymanbeachresort/videos/10158715980845230/